Tag: regression # How most statistical tests are linear models

Jonas Kristoffer Lindeløv wrote a great visual explanation of how the most common statistical tests (t-test, ANOVA, ANCOVA, etc) are all linear models in the back-end.

Jonas’ original blog uses R programming to visually show how the tests work, what the linear models look like, and how different approaches result in the same statistics.

George Ho later remade a Python programming version of the same visual explanation.

If I was thought statistics and methodology this way, I sure would have struggled less! Have a look yourself: https://lindeloev.github.io/tests-as-linear/ # Tutorial: Demystifying Deep Learning for Data Scientists

In this great tutorial for PyCon 2020, Eric Ma proposes a very simple framework for machine learning, consisting of only three elements:

1. Model
2. Loss function
3. Optimizer

By adjusting the three elements in this simple framework, you can build any type of machine learning program.

In the tutorial, Eric shows you how to implement this same framework in Python (using `jax`) and implement linear regression, logistic regression, and artificial neural networks all in the same way (using gradient descent).

I can’t even begin to explain it as well as Eric does himself, so I highly recommend you watch and code along with the Youtube tutorial (~1 hour):

If you want to code along, here’s the github repository: github.com/ericmjl/dl-workshop

Have you ever wondered what goes on behind the scenes of a deep learning framework? Or what is going on behind that pre-trained model that you took from Kaggle? Then this tutorial is for you! In this tutorial, we will demystify the internals of deep learning frameworks – in the process equipping us with foundational knowledge that lets us understand what is going on when we train and fit a deep learning model. By learning the foundations without a deep learning framework as a pedagogical crutch, you will walk away with foundational knowledge that will give you the confidence to implement any model you want in any framework you choose. # Animated Machine Learning Classifiers

Ryan Holbrook made awesome animated GIFs in R of several classifiers learning a decision rule boundary between two classes. Basically, what you see is a machine learning model in action, learning how to distinguish data of two classes, say cats and dogs, using some X and Y variables.

These visuals can be great to understand these algorithms, the models, and their learning process a bit better.

Here’s the original tweet, with the logistic regression animation. If you follow it, you will find a whole thread of classifier GIFs. These I extracted, pasted, and explained below.

Below is the GIF which I extracted using EZgif.com.

What you see is observations from two classes, say cats and dogs, each represented using colored dots. The dots are placed along X and Y axes, which represent variables about the observations. Their tail lengths and their hairyness, for instance.

Now there’s an optimal way to seperate these classes, which is the dashed line. That line best seperates the cats from the dogs based on these two variables X and Y. As this is an optimal boundary given this data, it is stable, it does not change.

However, there’s also a solid black line, which does change. This line represents the learned boundary by the machine learning model, in this case using logistic regression. As the model is shown more data, it learns, and the boundary is updated. This learned boundary represents the best line with which the model has learned to seperate cats from dogs.

Anything above the boundary is predicted to be class 1, a dog. Everything below predicted to be class 2, a cat. As logistic regression results in a linear model, the seperation boundary is very much linear/straight.

These animations are great to get a sense of how the models come to their boundaries in the back-end.

For instance, other machine learning models are able to use non-linear boundaries to dinstinguish classes, such as this quadratic discriminant analysis (qda). This “learned” boundary is much closer to the optimal boundary:

Models using multivariate adaptive regression splines (or MARS) seem to result in multiple linear boundaries pasted together:

Next, we have the k-nearest neighbors algorithm, which predicts for each point (animal) the class (cat/dog) based on the “k” points closest to it. As you see, this results in a highly fluctuating, localized boundary.

Now, Ryan decided to push the challenge, and simulate new data for two classes with a more difficult decision boundary. The new data and optimal boundaries look like this:

On these data, Ryan put a whole range of non-linear models to work.

Like this support-vector machine, which tries to create optimal boundaries built of support vectors around all the cats and all the dohs (this is definitely not a technical, error-free explanation of what’s happening here).

Generalized additive models are also cool to see in action. Why Ryan’s versions render so slowly, I don’t know. To learn more about GAMs, I strongly advise this tutorial here.

Let’s jump into some tree-based algorithms and the resulting models. A decision tree classifies data based on multiple, sequential, binary splits. Here, Ryan trained a simple decision tree:

As well as it’s big brother, a random forest, which uses hundreds of trees in the back end and thus results in a more flexible boundary:

Extreme gradient boosting is also a tree-based algorithm, which leverages many machine learning techniques to optimize the bias-variance tradeoff. Here’s an earlier blog on how to get started with Xgboost in Python or R:

Finally, a machine learning project is not complete without an artificial neural network. Learn more on these here:

If you want to know more about this project of Ryan Holbrook, do have a look at his accompanying blog here. You can also find Ryan’s code here on github. # Need to save R’s lm() or glm() models? Trim the fat!

I was training a predictive model for work for use in a Shiny App. However, as the training set was quite large (700k+ obs.), the model object to save was also quite large in size (500mb). This slows down your operation significantly!

Basically, all you really need are the coefficients (and a link function, in case of `glm()`). However, I can imagine that you are not eager to write new custom predictions functions, but that you would rather want to rely on R’s `predict.lm` and `predict.glm`. Hence, you’ll need to save some more object information.

Via Google I came to this blog, which provides this great custom R function (below) to decrease the object size of trained generalized linear models considerably! It retains only those object data that are necessary to make R’s predict functions work.

My saved linear model went from taking up half a GB to only 27kb! That’s a 99.995% reduction!

``````strip_glm = function(cm) {
cm\$y = c()
cm\$model = c()

cm\$residuals = c()
cm\$fitted.values = c()
cm\$effects = c()
cm\$qr\$qr = c()
cm\$linear.predictors = c()
cm\$weights = c()
cm\$prior.weights = c()
cm\$data = c()

cm\$family\$variance = c()
cm\$family\$dev.resids = c()
cm\$family\$aic = c()
cm\$family\$validmu = c()
cm\$family\$simulate = c()
attr(cm\$terms,".Environment") = c()
attr(cm\$formula,".Environment") = c()

cm
}`````` # Logistic regression is not fucked, by Jake Westfall

Recently, I came across a social science paper that had used linear probability regression. I had never heard of linear probability models (LPM), but it seems just an application of ordinary least squares regression but to a binomial dependent variable.

According to some, LPM is a commonly used alternative for logistic regression, which is what I was learned to use when the outcome is binary.

Potentially because of my own social science background (HRM), using linear regression without a link transformation on binary data just seems very unintuitive and error-prone to me. Hence, I sought for more information.

I particularly liked this article by Jake Westfall, which he dubbed “Logistic regression is not fucked”, following a series of blogs in which he talks about methods that are fucked and not useful.

Jake explains the classification problem and both methods inner workings in a very straightforward way, using great visual aids. He shows how LMP would differ from logistic models, and why its proposed benefits are actually not so beneficial. Maybe I’m in my bubble, but Jake’s arguments resonated.

http://jakewestfall.org/blog/index.php/2018/03/12/logistic-regression-is-not-fucked/

Here’s the summary:
Arguments against the use of logistic regression due to problems with “unobserved heterogeneity” proceed from two distinct sets of premises. The first argument points out that if the binary outcome arises from a latent continuous outcome and a threshold, then observed effects also reflect latent heteroskedasticity. This is true, but only relevant in cases where we actually care about an underlying continuous variable, which is not usually the case. The second argument points out that logistic regression coefficients are not collapsible over uncorrelated covariates, and claims that this precludes any substantive interpretation. On the contrary, we can interpret logistic regression coefficients perfectly well in the face of non-collapsibility by thinking clearly about the conditional probabilities they refer to. # StatQuest: Statistical concepts, clearly explained

Josh Starmer is assistant professor at the genetics department of the University of North Carolina at Chapel Hill.

But more importantly:
Josh is the mastermind behind StatQuest!

StatQuest is a Youtube channel (and website) dedicated to explaining complex statistical concepts — like data distributions, probability, or novel machine learning algorithms — in simple terms.

Once you watch one of Josh’s “Stat-Quests”, you immediately recognize the effort he put into this project. Using great visuals, a just-about-right pace, and relateable examples, Josh makes statistics accessible to everyone. For instance, take this series on logistic regression:

And do you really know what happens under the hood when you run a principal component analysis? After this video you will:

Or are you more interested in learning the fundamental concepts behind machine learning, then Josh has some videos for you, for instance on bias and variance or gradient descent:

With nearly 200 videos and counting, StatQuest is truly an amazing resource for students ‘and teachers on topics related to statistics and data analytics. For some of the concepts, Josh even posted videos running you through the analysis steps and results interpretation in the R language.

StatQuest started out as an attempt to explain statistics to my co-workers – who are all genetics researchers at UNC-Chapel Hill. They did these amazing experiments, but they didn’t always know what to do with the data they generated. That was my job. But I wanted them to understand that what I do isn’t magic – it’s actually quite simple. It only seems hard because it’s all wrapped up in confusing terminology and typically communicated using equations. I found that if I stripped away the terminology and communicated the concepts using pictures, it became easy to understand.

Over time I made more and more StatQuests and now it’s my passion on YouTube.